A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Yu Leng; Lampros Svolos; Dibyendu Adak; Ismael Boureima; Gianmarco Manzini; Hashem Mourad; Jeeyeon Plohr; Yu Leng; Lampros Svolos; Dibyendu Adak; Ismael Boureima; Gianmarco Manzini; Hashem Mourad; Jeeyeon Plohr

doi:10.3934/mine.2023100

Mathematics in Engineering

2023, Volume 5, Issue 6: 1-22. doi: 10.3934/mine.2023100

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Research article Special Issues

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

1.
T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
2.
T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
3.
XCP-5, Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Received: 13 August 2023 Revised: 21 October 2023 Accepted: 24 October 2023 Published: 09 November 2023

We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by ^[1,2,3], we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $C^0$ and $C^1$ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

Keywords:

Citation: Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, Jeeyeon Plohr. A guide to the design of the virtual element methods for second- and fourth-order partial differential equations[J]. Mathematics in Engineering, 2023, 5(6): 1-22. doi: 10.3934/mine.2023100

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Abstract

1. Introduction

Nonlinear oscillators have been widely used in various engineering and applied sciences, such as mathematics, physics, structural dynamics, mechanical engineering and other related fields of science ^{[1,2,3,4,5,6]}. Nonlinear differential equations (NDEs) can model many phenomena in various scientific aspects to present their effects and behaviors through mathematical principles. Perturbation methods are extremely beneficial when the nonlinear response is small ^[7,8,9,10]. In general, solving strongly nonlinear differential equations is very difficult. In ^[11], Mickens suggested an approximate expression for solving a truly nonlinear Duffing oscillator. Recently, various powerful analytical and numerical approximation techniques have been suggested for dealing with nonlinear oscillator differential equations. These include He's frequency-amplitude formulation ^[12], the harmonic balance method ^[13,14], the straightforward frequency prediction method ^[15], the modified harmonic balance method ^[16,17], the energy balance method ^[18,19,20], the homotopy perturbation method ^{[21,22,23,24,25]}, the Hamiltonian approach ^[26,27], the weighted averaging method ^[28], the global residue harmonic balance method ^[29,30,31], the max-min approach ^[32,33], Newton's harmonic balance method ^[34], the variational iteration method ^[35], the parameter-expansion method ^[36], the Lindstedt-Poincaré method ^[37,38] and the global error minimization method ^{[39,40,41,42]}.

The global error minimization method (GEMM) is one of the most frequently used techniques for dealing with the solutions of nonlinear oscillators, as it provides more accurate results valid for both weakly and strongly nonlinear oscillators than other known methods ^{[39,40,41,42]}. In the previous example, the solution up to the first approximation is calculated.

In this study, we extend and improve the global error minimization method up to a third order approximation to achieve the higher-order analytical solution of strongly nonlinear Duffing-harmonic oscillators. The present method is applied for two different problems, and the analytical results show that the modified global error minimization method (MGEMM) has better agreement with numerical solutions than the other analytical methods. Excellent agreement is observed between the approximate and exact solutions even for large amplitudes of the oscillation. Comparing the exact solutions with the approximate results has proved that the MGEMM is quite an accurate method in strongly nonlinear oscillator systems.

2. Basic idea of the modified global error minimization method (MGEMM)

To describe the proposed modified global error minimization method (MGEMM), we consider general second order nonlinear oscillator differential equations as follows:

$\ddot u + F(\dot u, \, u, \, t) = 0, \, \;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.\;\;\;\;$

(1)

We introducing $E(u)$ as a new function, defined as in ^[40,43], in the following form:

$E(u) = \int\nolimits_0^T \;{\left( {\ddot u + F(\dot u, \, u, \, t)} \right)^2}dt, \, \;\;\;\;\;T = 2\pi {\omega ^{ - 1}}.$

(2)

By assuming that $F(u)$ is an odd function, a general n-th order trial function of Eq (1) can be expressed as a sum of trigonometric functions as follows:

$u(t) = \mathop \sum \limits_{n = 0}^\infty ({a_{2n + 1)}}\cos ((2n + 1)\omega t)),$

(3)

where ${a_{(2n + 1)}}$ are unknown constant values which satisfy the relation

$A = \mathop \sum \limits_{n = 0}^\infty {a_{(2n + 1)}}.$

(4)

The following conditions were used to obtain the unknown parameters (i.e., ${a_{(2n + 1)}}$ and $\omega$ ):

$\frac{{\partial E(u)}}{{\partial \omega }} = 0, \, \;\;\;\;\;\;\;\;\frac{{\partial E(u)}}{{\partial {a_{(2n + 1)}}}} = 0, \, \;\;\;n \geqslant 1.\;\;$

(5)

By solving the n Eq (5) with the aid of Eq (4), the constants ${a_1}, \, \, {a_3}, {a_5}$ and the frequency of vibration $\omega$ are obtained.

3. Examples of two nonlinear Duffing-harmonic oscillators

In this section, two practical examples of nonlinear Duffing-harmonic oscillators are illustrated to show the effectiveness, accuracy and applicability of the proposed approach.

3.1. Example 1

In this application, we consider the following nonlinear Duffing-harmonic oscillator:

$\ddot u + {k_1}u + \frac{{{k_3}u}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0, \,$

(6)

where dots denote differentiation with respect to $t$ . Now, we shall study some different relevant cases considering Eq (6).

3.1.1. Case 1

First, we consider ${k_1} = 1$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Then, we have a nonlinear oscillator system having an irrational elastic item ^[12,16].

$\ddot u + u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0.$

(7)

According to the basic idea of the global error minimization method, the minimization problem of Eq (7) is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{u}{{1 + {u^2}}}} \right)^2}dt, \, \;\;\;T = 2\pi /\omega .$

(8)

The first-order approximate solution for Eq (7) can be represented as a trial function in the form

${u_1}(t) = {a_1}\cos (\omega t).$

(9)

Substituting Eq (9) into Eq (8) and choosing ${a_1} = A,$ it follows that

$E({u_1}) = \frac{{4{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{8\omega }} - 4{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{4}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(10)

Applying $\partial E({u_1})/\partial \omega = 0$ , the frequency of the nonlinear oscillator is obtained as follows:

$\omega = {\omega _1} = \sqrt {\frac{{16 + 18{A^2} + 5{A^4} + 2\sqrt {256 + 576{A^2} + 487{A^4} + 180{A^6} + 25{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(11)

In order to illustrate the capacity of the global error minimization method, the second-order approximation is applied to the Duffing-harmonic oscillator by using the following new trial solution:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t), \,$

(12)

where $A = {a_1} + {a_3}.$ By substituting Eq (12) into Eq (8), we have

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \tfrac{\pi }{{8\omega }}\left( {5{{({\omega ^2} - 1)}^2}a_1^6 + 5(11{\omega ^4} - 14{\omega ^2} + 3)a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4})} \right.} \\ { + \left. {3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + 3a_1^4\left( {4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2} \right)} \\ { + a_3^2\left( {8{{(2 - 9{\omega ^2})}^2} + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4} \right)} \\ { + \left. {a_1^2\left( {8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2 + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4} \right)} \right) = 0.} \end{array}} \right)\,$

(13)

Setting $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ leads to

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(20\omega ( - 1 + {\omega ^2})a_1^6 + 5( - 28\omega + 44{\omega ^3})a_1^5{a_3} + 2a_1^3{a_3}(8( - 18\omega + 20{\omega ^3}) + 3( - 100\omega + 436{\omega ^3})a_3^2} \\ { + 3a_1^4(4( - 6\omega + 4{\omega ^3}) + ( - 220\omega + 636{\omega ^3})a_3^2) + a_3^2( - 288\omega (2 - 9{\omega ^2}) + 12( - 54\omega + 324{\omega ^3})a_3^2} \\ { - 180\omega (1 - 9{\omega ^2})a_3^4) + a_1^2(32\omega ( - 2 + {\omega ^2}) + 16( - 90\omega + 236{\omega ^3})a_3^2 + 3( - 380\omega + 2236{\omega ^3})a_3^4))} \\ { - \tfrac{1}{{8{\omega ^2}}}\pi (5{{( - 1 + {\omega ^2})}^2}a_1^6 + 5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5{a_3} + 2a_1^3{a_3}(8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3\left( {5 - 50{\omega ^2}} \right.} \\ {\left. { + 109{\omega ^4}} \right)\, a_3^2) + 3a_1^4(4(2 - 3{\omega ^2} + {\omega ^4}) + (15 - 110{\omega ^2} + 159{\omega ^4})a_3^2) + a_3^2(8{{(2 - 9{\omega ^2})}^2}} \\ { + 12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^4) + a_1^2(8{{( - 2 + {\omega ^2})}^2} + 16(6 - 45{\omega ^2} + 59{\omega ^4})a_3^2} \\ { + 3(15 - 190{\omega ^2} + 559{\omega ^4})a_3^4)) = 0, } \end{array}} \right)\,$

(14)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}(5(3 - 14{\omega ^2} + 11{\omega ^4})a_1^5 + 6(15 - 110{\omega ^2} + 159{\omega ^4})a_1^4{a_3} + 12(5 - 50{\omega ^2} + 109{\omega ^4})a_1^3a_3^2} \\ { + 2a_1^3\left( {8(2 - 9{\omega ^2} + 5{\omega ^4}) + 3(5 - 50{\omega ^2} + 109{\omega ^4})a_3^2} \right) + a_3^2\left( {24(2 - 27{\omega ^2}81{\omega ^4}){a_3} \\+ 20{{(1 - 9{\omega ^2})}^2}a_3^3} \right)} \\ { + a_1^2\left( {32(6 - 45{\omega ^2} + 59{\omega ^4}){a_3} + 12(15 - 190{\omega ^2} + 559{\omega ^4})a_3^3} \right) + 2{a_3}\left( {8{{(2 - 9{\omega ^2})}^2}} \right.} \\ { + \left. {12(2 - 27{\omega ^2} + 81{\omega ^4})a_3^2 + 5{{(1 - 9{\omega ^2})}^2}a_3^2)} \right) = 0.} \end{array}} \right)\,$

(15)

For a known amplitude, the parameters of ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained by using the condition $A = {a_1} + {a_3}$ and solving Eqs (14) and (15). The computations were performed using the Mathematica software program, version 9.

To illustrate the capacity of this method, the third order approximation is applied by using the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t), \,$

(16)

where $A = {a_1} + {a_3} + {a_5}$ . Bringing Eq (16) into Eq (8) results in

$\left. \begin{gathered} \begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + a_3^2{{\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8\left( {2 - 9{\omega ^2}} \right){\omega ^2}} \right)}^2}} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \right.} \\ {\left. { + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right) + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \\ {4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(17)

Applying $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ yields

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {20a_1^6\omega \left( {{\omega ^2} - 1} \right) + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2a_1^5\omega \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 2a_1^3{a_3}\left( {3a_3^2\left( {436{\omega ^3} - 100\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 460\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right.} \\ {\left. { + 10a_5^2\left( {1004{\omega ^3} - 124\omega } \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 860\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {11576{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 52\omega } \right) + 8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 3a_5^2\left( {13596{\omega ^3} - 940\omega } \right)} \right) + a_5^2\left( {12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 800\omega \left( {2 - 25{\omega ^2}} \right)} \right.} \\ {\left. { - 500a_5^4\omega \left( {1 - 25{\omega ^3}} \right)} \right) + a_3^2\left( {3a_5^4\left( {22524{\omega ^3} - 1180\omega } \right) + 16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) \\- 288\omega \left( {2 - 9{\omega ^2}} \right)} \right)} \\ \begin{gathered} + a_1^4\left( {a_3^2\left( {1908{\omega ^3} - 660\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 380\omega } \right) + 3\left( {3a_5^2\left( {1108{\omega ^3} - 180\omega } \right) \\+ 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) \hfill \\ \begin{array}{*{20}{l}} {a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 380\omega } \right) + 18{a_5}a_3^3\left( {1364{\omega ^3} - 180\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) \\+ 32\omega \left( {{\omega ^2} - 2} \right)} \right.} \\ { + 9a_5^4\left( {5268{\omega ^3} - 340\omega } \right) + 4a_3^2\left( {3a_5^2\left( {6972{\omega ^3} - 700\omega } \right) + 4\left( {236{\omega ^3} - 90\omega } \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {10876{\omega ^3} - 860\omega } \right) + 8\left( {196{\omega ^3} - 54\omega } \right)} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 1} \right)}^2} + a_1^5\left( {{\omega ^2} - 1} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 3} \right) + 3{a_5}\left( {9{\omega ^2} - 1} \right)} \right) + 5a_3^6{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ { + 2a_1^3{a_3}\left( {3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) \\+ 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + a_5^2\left( {5a_3^4{{\left( {1 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) \\+ 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_3^2\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. {\left. { + 3a_5^2\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 8{{\left( {{\omega ^2} - 2} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. {\left. { + 6{a_3}{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \end{array}} \right)\,$

(18)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {5a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {11{\omega ^2} - 3} \right) + 30a_3^5{{\left( {1 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {2{a_3}\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right)} \right) + a_1^4\left( {2{a_3}\left( {477{\omega ^4} - 330{\omega ^2} + 45} \right) + 4{a_5}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {6{a_3}\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right) + 2a_1^3\left( {8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 3a_3^2\left( {109{\omega ^4} - 50{\omega ^2} + 5} \right) + 3{a_5}{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right) + 10a_5^2\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right)} \\ { + 12a_3^3\left( {a_5^2\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + \\ 4{a_1}{a_3}{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + 2{a_3}\left( {3a_5^4\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right) + 8{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 190{\omega ^2} + 15} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + \\ 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(19)

$\begin{gathered} \left. \begin{gathered} \begin{array}{*{20}{l}} {\, \tfrac{\pi }{{8\omega }}\left( {3a_1^5} \right.\left( {{\omega ^2} - 1} \right)\, \left( {9{\omega ^2} - 1} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 430{\omega ^2} + 15} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 230{\omega ^2} + 15} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 62{\omega ^2} + 3} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 190{\omega ^2} + 15} \right) + 18{a_5}\left( {277{\omega ^4} - 90{\omega ^2} + 5} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 6{a_5}\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right) + \\ 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ { + \left. {a_3^2\left( {2894{\omega ^4} - 620{\omega ^2} + 30} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 26{\omega ^2} + 1} \right) + 3a_5^2\left( {3399{\omega ^4} - 470{\omega ^2} + 15} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {1 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 590{\omega ^2} + 15} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 90{\omega ^2} + 5} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 36a_5^3\left( {1317{\omega ^4} - 170{\omega ^2} + 5} \right) + 12{a_3}a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 24a_3^2{a_5} \\\left( {1743{\omega ^4} - 350{\omega ^2} + 15} \right)} \\ {\left. {\left. { + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 430{\omega ^2} + 15} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right) \hfill \\ \, \hfill \\ \end{gathered}$

(20)

Now, by solving Eqs (18)–(20) and applying the condition $A = {a_1} + {a_3} + {a_5}$ , the parameters ${a_1}, \, {a_3}, \, {a_5}$ and the angular frequency $\omega$ can be obtained for the known amplitude $A,$ using the Mathematica software program, version 9. To examine the accuracy of the MGEMM solutions, the obtained results are compared with the frequency-amplitude formulation (FAF) ^[12], the energy balance method (EBM) ^[19], the modified harmonic balance method (MHBM) ^[16] and the exact solutions, as presented in Table 1 and Figure 1. We conclude that the third order approximation provides an excellent accuracy with respect to the exact numerical solutions.

Table 1. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{FAF}}$	${\omega _{EBM}}$	${\omega _{MHBM}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[12]	^[19]	^[16]	present	Exact
0.01	1.41419	1.41419	1.41419	1.41419	1.41419
0.1	1.41158	1.41158	1.41158	1.41158	1.41158
0.2	1.40388	1.40389	1.40390	1.40390	1.40390
0.4	1.37581	1.37595	1.37616	1.37616	1.37616
0.6	1.33694	1.33743	1.33827	1.33827	1.33827
0.8	1.29448	1.29550	1.29744	1.29743	1.29743
1	1.25375	1.25514	1.25845	1.25840	1.25842
5	1.02500	1.02588	1.03148	1.02945	1.03139
10	1.00656	1.00681	1.00895	1.00790	1.00893
100	1.00007	1.0000	1.00010	1.00010	1.00010
1000	1.00000	1.00000	1.00000	1.00000	1.00000

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Figure 1. Comparison of the approximate solution (red line) with the numerical solution (blue line).

DownLoad: Full-Size Img PowerPoint

3.1.2. Case 2

Now, if we put ${k_1} = 0$ , ${k_2} = 1$ and ${k_3} = 1$ in Eq (6), we obtain the following nonlinear oscillator, in which the restoring force has a rational expression ^[21,26].

$\ddot u + \frac{u}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(21)

Using the previously mentioned procedure, the solution up to a third-order approximation is calculated. Depending on the analytical approximation, first, second or third, the approximate solution is assumed in the forms of (9), (12) and (16), respectively.

Finally, as in Case 1, the third order approximate solutions are compared with the homotopy perturbation method (HPM) ^[21], the Hamiltonian approach (HA) ^[26] and the exact solutions, as displayed in Table 2.

Table 2. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{HPM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[21]	^[26]	present	Exact
0.01	0.999963	0.9999625	0.999963	0.99999
0.1	0.996271	0.99627403	0.996273	0.9991208
1	0.755929	0.765366864	0.761539	0.76157808
10	0.114708	0.13420106	0.11948	0.123322
100	0.0115462	0.01407125	0.0120357	0.0125265

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3.2. Example 2

In the second application, we will consider the following nonlinear Duffing-harmonic oscillator ^[18]:

$\ddot u + {k_1}u + \frac{{{k_3}{u^3}}}{{1 + {k_2}{u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\;\;\dot u(0) = 0, \,$

(22)

where dots denote differentiation with respect to $t$ . Now, we consider some following cases to compare the present solutions with published solutions using different approximate analytical methods.

3.2.1. Case 1

First, we consider ${k_1} = 1,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (22). Then, we have the following nonlinear Duffing-harmonic equation ^[18,28]:

$\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \, \, \dot u(0) = 0.$

(23)

The minimization problem is

$E(u) = \int\nolimits_0^T {\left( {\ddot u + u + \frac{{{u^3}}}{{1 + {u^2}}}} \right)^2}dt, \, \;\;T = \frac{{2\pi }}{\omega }.$

(24)

For the first-order approximation, assume that the trial function is given by

${u_1}(t) = {a_1}\cos (\omega t), \,$

(25)

where $A = {a_1}$ . By inserting Eq (25) into Eq (24), we obtain

$E({u_1}) = \frac{{{A^2}\pi }}{\omega } + \frac{{3{A^4}\pi }}{\omega } + \frac{{5{A^6}\pi }}{{2\omega }} - 2{A^2}\pi \omega - \frac{9}{2}{A^4}\pi \omega - \frac{5}{2}{A^6}\pi \omega + {A^2}\pi {\omega ^3} + \frac{3}{2}{A^4}\pi {\omega ^3} + \frac{5}{8}{A^6}\pi {\omega ^3} = 0.$

(26)

The frequency can be found through the condition $\partial E({u_1})/\partial \omega = 0$ , as follows:

$\omega = {\omega _1} = \sqrt {\frac{{8 + 18{A^2} + 10{A^4} + 2\sqrt {64 + 288{A^2} + 487{A^4} + 360{A^6} + 100{A^8}} }}{{3(8 + 12{A^2} + 5{A^4})}}} .$

(27)

To improve the analytical approximation, we add additional terms to the trial function:

${u_2}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t).$

(28)

The constraint of this minimization is $A = {a_1} + {a_3}$ . Substituting the above new trial function into Eq (24), we obtain

$\left. {\begin{array}{*{20}{l}} {E({u_2}) = \, \tfrac{\pi }{{8\omega }}\left( {5a_1^6} \right.{{\left( {{\omega ^2} - 2} \right)}^2} + 5a_1^5{a_3}\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \\ { + a_3^2\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \\ {\left. { + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(29)

By using $\partial E({u_2})/\partial \omega = 0$ and $\partial E({u_2})/\partial {a_3} = 0$ , it follows that

$\left. {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5{a_3}a_1^5} \right.\, \left( {44{\omega ^3} - 56\omega } \right) + 3a_1^4\left( {a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right) + 20a_1^6\omega \left( {{\omega ^2} - 2} \right)} \end{array}} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 8\left( {20{\omega ^3} - 18\omega } \right)} \right) + a_3^2\left( {12a_3^2\left( {324{\omega ^3} - 54\omega } \right) - 288\omega \left( {1 - 9{\omega ^3}} \right)} \right.} \\ {\left. {\left. { - 180a_3^4\omega \left( {2 - 9{\omega ^3}} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 16a_3^2\left( {236{\omega ^3} - 90\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right)} \right)} \\ { - \tfrac{\pi }{{8{\omega ^2}}}\left( {5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + 5{a_3}a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 3a_1^4\left( {a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right)} \right.} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \\ {\left. {\left. { + 5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 16a_3^2\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right)} \right) = 0, } \end{array}} \right)\,$

(30)

$\left. {\begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\left( {5a_1^5\left( {11{\omega ^4} - 28{\omega ^2} + 12} \right) + 6{a_3}a_1^4\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4a_3^2a_1^3\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \\ { + 2a_1^3\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 8\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right) + a_3^2\left( {24{a_3}\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + 20a_3^3{{\left( {2 - 9{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 32{a_3}\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right)} \\ {\left. { + 2{a_3}\left( {5a_3^4{{\left( {2 - 9{\omega ^2}} \right)}^2} + 12a_3^2\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right) + 8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right)} \right) = 0.} \end{array}} \right)\,$

(31)

The minimization problem's conditions can be easily achieved by replacing ${a_1} = A - {a_3}$ , and the parameters ${a_1}$ , ${a_3}$ and angular frequency $\omega$ can be obtained for a known amplitude $A$ .

To show the accuracy of the MGEM method in higher order approximations, we apply the third order approximation and consider the following trial function:

${u_3}(t) = {a_1}\cos (\omega t) + {a_3}\cos (3\omega t) + {a_5}\cos (5\omega t).$

(32)

Using Eq (32) as the trial function in Eq (24), where $A = {a_1} + {a_3} + {a_5}$ , leads to

$\left. {\begin{array}{*{20}{l}} {E({u_3}) = \, \tfrac{{\pi \omega }}{8}\left( {5a_1^6} \right.\left( {{\omega ^2} - 4} \right) + a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 28} \right) + 3{a_5}\left( {9{\omega ^2} - 20} \right)} \right) + 45a_3^6\left( {9{\omega ^2} - 4} \right)} \\ { + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^2} - 300} \right) + 10a_5^2\left( {251{\omega ^2} - 124} \right) + 3{a_3}{a_5}\left( {663{\omega ^2} - 460} \right) + 40{\omega ^2}} \right)} \\ { + a_1^4\left( {a_3^2\left( {477{\omega ^2} - 660} \right) + 9a_5^2\left( {277{\omega ^2} - 180} \right) + 4{a_3}{a_5}\left( {367{\omega ^2} - 380} \right) + 12{\omega ^2}} \right)} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^2} - 860} \right) + 324{\omega ^2}} \right) + 2{a_1}a_3^2{a_5}\left( {2a_3^2\left( {1447{\omega ^2} - 620} \right) + 2296{\omega ^2}} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^2} - 940} \right) + 15{a_3}{a_5}\left( {153{\omega ^2} - 52} \right)} \right) + 125a_5^2\left( {60a_5^2{\omega ^2} + 40{\omega ^2}} \right.} \\ {\left. { + a_5^4\left( {25{\omega ^2} - 4} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^2} - 380} \right) + 944a_3^2{\omega ^2} + 8{\omega ^2}} \right.} \\ { + 6{a_3}{a_5}\left( {3a_3^2\left( {341{\omega ^2} - 180} \right) + 392{\omega ^2}} \right) + 12a_5^2\left( {7a_3^2\left( {249{\omega ^2} - 100} \right) + 484{\omega ^2}} \right)} \\ {\left. { + 9a_5^4\left( {1317{\omega ^2} - 340} \right) + 6{a_3}a_5^3\left( {2719{\omega ^2} - 860} \right)} \right) + a_3^2\left( {12848a_5^2{\omega ^2} + 648{\omega ^2}} \right.} \\ {\left. {\left. { + 3a_5^4\left( {5631{\omega ^2} - 1180} \right)} \right)} \right) = 0.} \end{array}} \right)\,$

(33)

By setting $\partial E({u_3})/\partial \omega = 0$ , $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ , we obtain

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {20a_1^6\omega } \right.\left( {{\omega ^2} - 2} \right) - 180a_3^6\omega \left( {2 - 9{\omega ^2}} \right) + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {110{a_3}\omega + 54{a_5}\omega } \right) + 2{\omega _3}a_1^5\left( {5{a_3}\left( {11{\omega ^2} - 6} \right)} \right.} \\ {\left. { + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 3a_3^4\left( {a_5^2\left( {11644{\omega ^3} - 1720\omega } \right) + 4\left( {324{\omega ^3} - 54\omega } \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {1148{\omega ^3} - 198\omega } \right)} \right.} \\ {\left. { + 2a_3^2\left( {5788{\omega ^3} - 1240\omega } \right) + 15{a_5}{a_3}\left( {612{\omega ^3} - 104\omega } \right) + 3a_5^2\left( {13596{\omega ^3} - 1880\omega } \right)} \right) + a_5^2\left( { - 800\omega \left( {1 - 25{\omega ^2}} \right)} \right.} \\ {\left. { + 12a_5^2\left( {2500{\omega ^3} - 150\omega } \right) - 500a_5^4\omega \left( {2 - 25{\omega ^2}} \right)} \right) + a_3^2\left( {16a_5^2\left( {3212{\omega ^3} - 306\omega } \right) - 288\omega \left( {1 - 9{\omega ^2}} \right)} \right.} \\ {\left. { + 3a_5^4\left( {22524{\omega ^3} - 2360\omega } \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {1308{\omega ^3} - 600\omega } \right) + 3{a_5}{a_3}\left( {2652{\omega ^3} - 920\omega } \right)} \right.} \\ {\left. { + 2\left( {5a_5^2\left( {1004{\omega ^3} - 248\omega } \right) + 4\left( {20{\omega ^3} - 18\omega } \right)} \right)} \right) + a_1^4\left( {3a_3^2\left( {636{\omega ^3} - 440\omega } \right) + 4{a_5}{a_3}\left( {1468{\omega ^3} - 760\omega } \right)} \right.} \\ {\left. { + 3\left( {a_5^2\left( {3324{\omega ^3} - 1080\omega } \right) + 4\left( {4{\omega ^3} - 6\omega } \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {2236{\omega ^3} - 760\omega } \right) + 32\omega \left( {{\omega ^2} - 1} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ { + 18a_3^3{a_5}\left( {1364{\omega ^3} - 360\omega } \right) + 48a_5^2\left( {484{\omega ^3} - 78\omega } \right) + 9a_5^4\left( {5268{\omega ^3} - 680\omega } \right) + 4a_3^2\left( {4\left( {236{\omega ^3} - 90\omega } \right)} \right.} \\ {\left. { - \tfrac{{1 - \pi }}{{8{\omega ^2}}}} \right)\, 5a_1^6{{\left( {{\omega ^2} - 2} \right)}^2} + a_1^5\left( {{\omega ^2} - 2} \right)\, \left( {5{a_3}\left( {11{\omega ^2} - 6} \right) + 3{a_5}\left( {9{\omega ^2} - 2} \right)} \right) + 5a_3^6{{\left( {2 - 9{\omega ^2}} \right)}^2}} \\ { + 3a_3^4\left( {a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right) + 2{a_1}a_3^2{a_5}\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2} + 12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right) + a_3^2\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right. + } \\ {\left. {3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3{a_3}\left( {a_3^2\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^4\left( {3a_3^2\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right. + 3\left( {4\left( {{\omega ^4} - 3{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {a_5^2 + \left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \right) + a_1^2\left( {3a_3^4\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 8{{\left( {{\omega ^2} - 1} \right)}^2}} \right.} \\ { + 9a_5^4\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 48a_5^2\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right) + 18a_3^3{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right)} \\ { + 4a_3^2\left( {3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right) + 6{a_3}{a_5}\left( {8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. {\left. {\left. { + a_5^2 + \left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(34)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{\pi }{{8\omega }}\, \left( {3a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {9{\omega ^2} - 2} \right) + 6a_3^4{a_5}\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right) + 2a_1^3{a_3}\left( {3{a_3}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 20{a_5}\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right)} \right) + a_1^4\left( {4{a_3}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right) + 6{a_5}\left( {831{\omega ^4} - 540{\omega ^2} + 60} \right)} \right)} \\ { + 2{a_1}a_3^2{a_5}\left( {15{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 6{a_5}\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right) + 2{a_1}a_3^2\left( {8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 15{a_5}{a_3}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right) + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right)} \right)} \\ { + a_5^2\left( {20a_5^3{{\left( {2 - 25{\omega ^2}} \right)}^2} + 24{a_5}\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right)} \right) + a_3^2\left( {32{a_5}\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 12a_5^3\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right)} \right) + 2{a_5}\left( {12a_5^2\left( {625{\omega ^4} - 75{\omega ^2} + 2} \right) + 8{{\left( {1 - 25{\omega ^2}} \right)}^2}} \right.} \\ {\left. { + 5a_5^4{{\left( {2 - 25{\omega ^2}} \right)}^2}} \right) + a_1^2\left( {18a_3^3\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + 96{a_5}\left( {121{\omega ^4} - 39{\omega ^2} + 2} \right)} \right.} \\ { + 24{a_5}a_3^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right) + 12a_5^2{a_3}\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right)} \\ {\left. {\left. { + 36a_5^3\left( {1317{\omega ^4} - 340{\omega ^2} + 20} \right) + 6{a_3}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0, } \end{array} \hfill \\ \end{gathered} \right)\,$

(35)

$\left. \begin{gathered} \begin{array}{*{20}{l}} {\tfrac{1}{{8\omega }}\pi \left( {\, 5a_1^5} \right.\left( {{\omega ^2} - 2} \right)\, \left( {11{\omega ^2} - 6} \right) + 30a_3^5{{\left( {2 - 9{\omega ^2}} \right)}^2} + 2{a_1}a_3^2{a_5}\left( {4{a_3}\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 15{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + a_1^4\left( {6{a_3}\left( {159{\omega ^4} - 220{\omega ^2} + 60} \right) + 4{a_5}\left( {367{\omega ^4} - 380{\omega ^2} + 60} \right)} \right)} \\ { + 2a_1^3{a_3}\left( {2{a_3}\left( {327{\omega ^4} - 300{\omega ^2} + 60} \right) + 3{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right)} \right) + 12a_3^3\left( {4\left( {81{\omega ^4} - 27{\omega ^2} + 2} \right)} \right.} \\ {\left. { + a_5^2\left( {2911{\omega ^4} - 860{\omega ^2} + 60} \right)} \right) + 4{a_1}{a_3}{a_5}\left( {2a_3^2\left( {1447{\omega ^4} - 620{\omega ^2} + 60} \right) + 8\left( {287{\omega ^4} - 99{\omega ^2} + 6} \right)} \right.} \\ {\left. { + 3a_5^2\left( {3399{\omega ^4} - 940{\omega ^2} + 60} \right) + 15{a_3}{a_5}\left( {153{\omega ^4} - 52{\omega ^2} + 4} \right)} \right) + 2{a_3}\left( {8{{\left( {1 - 9{\omega ^2}} \right)}^2}} \right.} \end{array} \hfill \\ \begin{array}{*{20}{l}} {\left. { + 3a_5^4\left( {5631{\omega ^4} - 1180{\omega ^2} + 60} \right) + 16a_5^2\left( {803{\omega ^4} - 153{\omega ^2} + 6} \right)} \right) + 2a_1^3\left( {a_3^2\left( {327\omega _3^4 - 300{\omega ^2} + 60} \right)} \right.} \\ {\left. { + 3{a_3}{a_5}\left( {663{\omega ^4} - 460{\omega ^2} + 60} \right) + 2\left( {5a_5^2\left( {251{\omega ^4} - 124{\omega ^2} + 12} \right) + 4\left( {5{\omega ^4} - 9{\omega ^2} + 2} \right)} \right)} \right)} \\ { + a_1^2\left( {12a_3^3\left( {559{\omega ^4} - 380{\omega ^2} + 60} \right) + 54a_3^2{a_5}\left( {341{\omega ^4} - 180{\omega ^2} + 20} \right) + } \right.\, 8{a_3}\left( {4\left( {59{\omega ^4} - 45{\omega ^2} + 6} \right)} \right.} \\ {\left. {\left. {\left. { + 3a_5^2\left( {1743{\omega ^4} - 700{\omega ^2} + 60} \right)} \right) + 6{a_5}\left( {a_5^2\left( {2719{\omega ^4} - 860{\omega ^2} + 60} \right) + 8\left( {49{\omega ^4} - 27{\omega ^2} + 2} \right)} \right)} \right)} \right) = 0.} \end{array} \hfill \\ \end{gathered} \right)\,$

(36)

Putting ${a_1} = A - {a_3} - {a_5}$ in the minimization problem changes the constraint minimization problem to an unconstrained minimization problem, which is easier to find the solution of Eq (33) by using the conditions $\partial E({u_3})/\partial \omega = 0,$ $\partial E({u_3})/\partial {a_3} = 0$ and $\partial E({u_3})/\partial {a_5} = 0$ .

We plot the analytical solutions obtained from MGEMM (red line) Eq (32) and compare them with numerical solutions of Eq (23) obtained using the fourth order Runge-Kutta method (blue line). It is observed that for all different values of the amplitude A, the approximate solutions match extremely well with the numerical solutions (see Figure 2).

Figure 2. Comparison of the approximate solution (red line) with the numerical solution (blue line).

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3.2.2. Case 2

Second, we consider ${k_1} = 0,$ ${k_2} = 1$ and ${k_3} = 1$ in Eq (6). Hence, we have the one-dimensional nonlinear oscillator governed by ^{[21,22,23,24,25,26]}

$\ddot u + \frac{{{u^3}}}{{1 + {u^2}}} = 0, \, \;\;\;\;\;u(0) = A, \, \;\;\;\dot u(0) = 0.$

(37)

Finally, we can obtain the first and second-order approximations to Eq (37) given by Eqs (25) and (28), respectively. We remark that the third-order approximation in Eq (32) can be given by MGEMM in a similar manner. Generally, after three steps of MGEMM, one can obtain the approximated solutions to Eq (37) with sufficient accuracy. The analytical results of Eq (37) are compared with the iterative homotopy harmonic balance method (IHHBM) ^[34], the energy balance method (EBM) ^[20], the max-min approach (MMA) ^[32], the global residue harmonic balance method (GRHBM) ^[29], the Hamiltonian approach (HA) ^[26] and the exact solutions, as shown in Table 3.

Table 3. Comparison of the approximate analytical frequencies with the exact solutions.

$A$	${\omega _{EBM}}$	${\omega _{IHHBM}}$	${\omega _{MMA}}$	${\omega _{GRHBM}}$	${\omega _{HA}}$	${\omega _{3rd\, GEMM}}$	${\omega _{exact}}$
	^[20]	^[34]	^[32]	^[29]	^[26]	present	Exact
0.01	0.00866	0.008478	0.00866	0.008472	0.00865	0.00847	0.00847
0.1	0.08627	0.084418	0.08627	0.084394	0.08624	0.08439	0.08439
1	0.65164	0.63136	0.65465	0.636795	0.64359	0.636783	0.63678
5	0.97343	0.96667	0.97435	0.968107	0.96731	0.969202	0.96698
10	0.99314	0.99090	0.99340	0.991591	0.99095	0.992005	0.99092
50	0.99973	0.99961	0.99973	0.999657	0.999608	0.999676	0.99961
100	0.99999	0.999901	0.99993	0.999914	0.99990	0.999919	0.99990

| Show Table

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4. Results and discussion

In this paper, we test the analytical solutions of strongly nonlinear Duffing-harmonic oscillators to show the effectiveness of MGEMM. Comparisons of the analytical solutions and the exact numerical solutions of the Duffing-harmonic oscillators for small and large values of the amplitude have been illustrated in Figures 1 and 2 and Tables 1–3. In these Figures, the analytical solution is indicated by a red line, while the numerical solution is represented by a blue line. There is good compatibility between the analytical and numerical solutions, which confirms the accuracy of our results, and excellent matching is observed in these calculations. Moreover, as shown in Tables 1–3, the results produced using the MGEMM are in better agreement with those obtained using exact solutions than other existing ones in the literature. The calculations of the above applications were done using Mathematica software.

5. Conclusions

In this paper, the modified global error minimization method has been presented successfully to obtain higher order approximate periodic solutions of strongly nonlinear Duffing-harmonic oscillators. The present method, which is proved to be a powerful mathematical tool to study nonlinear oscillators, can be easily extended to any nonlinear equation because of its efficiency and convenient applicability. We demonstrated the accuracy and efficiency of the proposed method by solving some examples. We showed that the obtained solutions are valid for the whole domain. Comparisons of the obtained solutions, whether numerical or analytical, revealed a clear match, highlighting the precision of the modified GEMM.

Acknowledgments

M. Zayed and G. M. Ismail extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through the research groups program under grant R.G.P.2/207/43.

Conflict of interest

The authors declare that they have no conflict of interest.

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Mathematics in Engineering

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Related Papers:

Abstract

1. Introduction

2. Basic idea of the modified global error minimization method (MGEMM)

3. Examples of two nonlinear Duffing-harmonic oscillators

3.1. Example 1

3.1.1. Case 1

3.1.2. Case 2

3.2. Example 2

3.2.1. Case 1

3.2.2. Case 2

4. Results and discussion

5. Conclusions

Acknowledgments

Conflict of interest

References

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Mathematics in Engineering

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Related Papers:

Abstract

1. Introduction

2. Basic idea of the modified global error minimization method (MGEMM)

3. Examples of two nonlinear Duffing-harmonic oscillators

3.1. Example 1

3.1.1. Case 1

3.1.2. Case 2

3.2. Example 2

3.2.1. Case 1

3.2.2. Case 2

4. Results and discussion

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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